Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27×C4 |
| direct product | C3×#42995 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2,3)(5,7)(8,9,11,13)(10,14,15,12)(16,17,21,20)(18,24,26,22)(19,27,29,23)(25,30,28,31);(8,10,11,15)(9,12,13,14)(17,20)(22,24)(23,27)(30,31);(4,5)(6,7)(8,10)(9,12)(11,15)(13,14)(16,18,21,26)(17,22,20,24)(19,28,29,25)(23,31,27,30);(8,9)(10,12)(11,13)(14,15)(16,19,21,29)(17,23,20,27)(18,25,26,28)(22,30,24,31);(16,20,21,17)(18,24,26,22)(19,27,29,23)(25,31,28,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35983);